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## G = C52⋊Q8order 200 = 23·52

### The semidirect product of C52 and Q8 acting faithfully

Aliases: C52⋊Q8, C52⋊C4.C2, C5⋊D5.3C22, SmallGroup(200,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5⋊D5 — C52⋊Q8
 Chief series C1 — C52 — C5⋊D5 — C52⋊C4 — C52⋊Q8
 Lower central C52 — C5⋊D5 — C52⋊Q8
 Upper central C1

Generators and relations for C52⋊Q8
G = < a,b,c,d | a5=b5=c4=1, d2=c2, ab=ba, cac-1=b-1, dad-1=a3, cbc-1=a, dbd-1=b2, dcd-1=c-1 >

25C2
2C5
2C5
2C5
25C4
25C4
25C4
10D5
10D5
10D5
25Q8
10F5
10F5
10F5

Character table of C52⋊Q8

 class 1 2 4A 4B 4C 5A 5B 5C size 1 25 50 50 50 8 8 8 ρ1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 1 1 1 1 linear of order 2 ρ3 1 1 1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 -1 1 -1 1 1 1 linear of order 2 ρ5 2 -2 0 0 0 2 2 2 symplectic lifted from Q8, Schur index 2 ρ6 8 0 0 0 0 -2 3 -2 orthogonal faithful ρ7 8 0 0 0 0 3 -2 -2 orthogonal faithful ρ8 8 0 0 0 0 -2 -2 3 orthogonal faithful

Permutation representations of C52⋊Q8
On 10 points - transitive group 10T20
Generators in S10
```(1 2 3 4 5)(6 7 8 9 10)
(1 3 5 2 4)(6 9 7 10 8)
(2 3 5 4)(6 10 7 8)
(1 9)(2 6 5 7)(3 8 4 10)```

`G:=sub<Sym(10)| (1,2,3,4,5)(6,7,8,9,10), (1,3,5,2,4)(6,9,7,10,8), (2,3,5,4)(6,10,7,8), (1,9)(2,6,5,7)(3,8,4,10)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10), (1,3,5,2,4)(6,9,7,10,8), (2,3,5,4)(6,10,7,8), (1,9)(2,6,5,7)(3,8,4,10) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10)], [(1,3,5,2,4),(6,9,7,10,8)], [(2,3,5,4),(6,10,7,8)], [(1,9),(2,6,5,7),(3,8,4,10)]])`

`G:=TransitiveGroup(10,20);`

On 20 points - transitive group 20T47
Generators in S20
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 9)(2 6 5 7)(3 8 4 10)(11 18 12 16)(13 19 15 20)(14 17)
(1 14)(2 11 5 12)(3 13 4 15)(6 16 7 18)(8 20 10 19)(9 17)```

`G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,16)(13,19,15,20)(14,17), (1,14)(2,11,5,12)(3,13,4,15)(6,16,7,18)(8,20,10,19)(9,17)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,9)(2,6,5,7)(3,8,4,10)(11,18,12,16)(13,19,15,20)(14,17), (1,14)(2,11,5,12)(3,13,4,15)(6,16,7,18)(8,20,10,19)(9,17) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,9),(2,6,5,7),(3,8,4,10),(11,18,12,16),(13,19,15,20),(14,17)], [(1,14),(2,11,5,12),(3,13,4,15),(6,16,7,18),(8,20,10,19),(9,17)]])`

`G:=TransitiveGroup(20,47);`

On 25 points: primitive - transitive group 25T17
Generators in S25
```(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)
(1 25 9 12 19)(2 21 10 13 20)(3 22 6 14 16)(4 23 7 15 17)(5 24 8 11 18)
(2 25 5 19)(3 9 4 12)(6 7 15 14)(8 17 13 22)(10 23 11 16)(18 20 21 24)
(2 3 5 4)(6 24 15 20)(7 21 14 18)(8 23 13 16)(9 25 12 19)(10 22 11 17)```

`G:=sub<Sym(25)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,25,9,12,19)(2,21,10,13,20)(3,22,6,14,16)(4,23,7,15,17)(5,24,8,11,18), (2,25,5,19)(3,9,4,12)(6,7,15,14)(8,17,13,22)(10,23,11,16)(18,20,21,24), (2,3,5,4)(6,24,15,20)(7,21,14,18)(8,23,13,16)(9,25,12,19)(10,22,11,17)>;`

`G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25), (1,25,9,12,19)(2,21,10,13,20)(3,22,6,14,16)(4,23,7,15,17)(5,24,8,11,18), (2,25,5,19)(3,9,4,12)(6,7,15,14)(8,17,13,22)(10,23,11,16)(18,20,21,24), (2,3,5,4)(6,24,15,20)(7,21,14,18)(8,23,13,16)(9,25,12,19)(10,22,11,17) );`

`G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25)], [(1,25,9,12,19),(2,21,10,13,20),(3,22,6,14,16),(4,23,7,15,17),(5,24,8,11,18)], [(2,25,5,19),(3,9,4,12),(6,7,15,14),(8,17,13,22),(10,23,11,16),(18,20,21,24)], [(2,3,5,4),(6,24,15,20),(7,21,14,18),(8,23,13,16),(9,25,12,19),(10,22,11,17)]])`

`G:=TransitiveGroup(25,17);`

C52⋊Q8 is a maximal subgroup of   D5≀C2⋊C2
C52⋊Q8 is a maximal quotient of   (C5×C10).Q8

Polynomial with Galois group C52⋊Q8 over ℚ
actionf(x)Disc(f)
10T20x10-2x9-101x8+272x7+691x6-3142x5+3299x4+72x3-1939x2+978x-125233·37·58·234·292·1092·1272·12832

Matrix representation of C52⋊Q8 in GL8(ℤ)

 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1
,
 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 -1 -1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 -1 -1 -1 -1

`G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1],[0,0,0,-1,0,0,0,0,1,0,0,-1,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,1,0,0,0,0,0,-1,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,-1] >;`

C52⋊Q8 in GAP, Magma, Sage, TeX

`C_5^2\rtimes Q_8`
`% in TeX`

`G:=Group("C5^2:Q8");`
`// GroupNames label`

`G:=SmallGroup(200,44);`
`// by ID`

`G=gap.SmallGroup(200,44);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-5,5,20,61,26,323,408,173,3004,409,1014]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^5=b^5=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=b^-1,d*a*d^-1=a^3,c*b*c^-1=a,d*b*d^-1=b^2,d*c*d^-1=c^-1>;`
`// generators/relations`

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